Questions tagged [projective-geometry]

Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
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Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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Resolutions in Algebraic Geometry

I guess that the answer to this question can be given using Gröbner basis among many other computational methods, but my goal is to see if there is a more elementary way of approaching this problem. ...
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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The geometry of $\operatorname{PSO}(4)$ and the quaternions

Question: Given a twist of the projective space, how do I find unit quaternions that represent it? Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
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Finite projective plane order 11

Consider a finite projective plane of order 11, which is there more of: a) Unordered 4-tuples of lines with a non-empty intersection? b) Unordered 7-tuples of points which belong to the same line? ...
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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Connection between number theory and projective geometry?

Consider the directed graphs $G_n^p$ with node set $\{0,1,\dots,p-1\}$, $0 \leq n < p$ with an arrow from $a$ to $b$ if $na=b\operatorname{mod}p$. These graphs are graphical multiplication tables ...
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Calculating singular points of a quintic curve in the projective plane?

I have the following question for an assignment. "An irreducible quintic curve in the real projective plane $P^2(R)$ is defined by $F: (X^2-Z^2)^2Y-(Y^2-Z^2)^2X=0$ Verify that the quartic curve ...
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Lines lying in a projective algebraic set

I want to find all lines lying entirely in the projective algebraic set $XY-ZW=0$ in $\mathbb{P}^3$, where $X, Y, Z, W$ are homogeneous coordinates. How can I do this?
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A question on Grassmannian

Let $T: Gr(n,\mathbb C^{2n}) \rightarrow G(n,\mathbb C^{2n})$ be the involution defined by $W \rightarrow W^{\perp}$ with respect to a symplectic form on $\mathbb C^{2n}$. Is there a direct proof (...
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$O$ is the point inside triangle $ABC$ . The lines joining the three vertices $A, B, C$ to $O$ cut the opposite sides in $K, L$, and $M$ respectively. A line through $M$ parallel to $KL$ cuts the ...
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